Vol. 53, No. 2, 1974

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Characterization of a function by certain infinite series it generates

Charles Kam-Tai Chui and Philip Wesley Smith

Vol. 53 (1974), No. 2, 363–371

Let A be a set of real numbers and F be a class of complex-valued functions defined on the real line such that for each f F the infinite series S(x,f) = k=1f(kαj) converges for every nonzero x in A. If 0 A, we set S(0,f) = f(0). It seems to be an interesting problem to study the different sets A and function classes F such that each f F is uniquely determined by the sums S(x,f) where x A. Clearly, the larger the class F is studied, the larger set A is needed to guarantee uniqueness. We have positive results for a class of entire functions of exponential type and for fairly large classes of continuous functions. Some examples are also given to show that in general A cannot be too small.

Mathematical Subject Classification
Primary: 30A66
Secondary: 42A68
Received: 25 March 1973
Published: 1 August 1974
Charles Kam-Tai Chui
Philip Wesley Smith